1. Topic 4 - Continuous distributions
Basics of continuous distributions - pages
Uniform distribution - pages 135 – 136
Normal distribution - pages
Gamma distribution - pages

2. Continuous Random Variables
A continuous random variable can take on values from an entire interval of the real line.
The probability density function (pdf) of a continuous random variable, X, is a function f(x) such that for a < b
The cdf of X is defined as

3. Some relationships What is the relationship between f and F?
P(a ≤ X ≤ b) = F(b) – F(a)
P(X = a) = P(a ≤ X ≤ a) = F(a) – F(a) = 0

4. Pipeline example
A pipeline is 100 miles long and every location along the pipeline is equally likely to break
Let X be the distance measured in miles from the pipeline origin where a break occurs
What is the cdf for X?
What is the pdf for X?
What is P(30 ≤ X ≤ 50)?

5. Requirements of a pdf
A pdf must satisfy the following two requirements:
Does the pipeline pdf satisfy these requirements?

6. Uniform distribution
A uniform distribution on the interval from A to B, U(A,B), is defined by a pdf of the form
Does f(x) meet requirements?
What is the cdf for the Uniform distribution?

7. Mean and variance of a cont. random variable

8. Back to the Uniform What is the mean of a U(0,1) distribution?
What is the variance of U(0,1) distribution?

9. Gamma distribution
The gamma distribution, G(a,b), is defined by the following pdf
where
Properties of the gamma function, G(a)
For a > 1, G(a) = (a-1)G(a-1)
If a is a positive integer, G(a) = (a-1)!
G(1/2) =

10. Properties of the gamma distribution
Is it a valid pdf?
Show
Show m = ab

11. More on the gamma distribution
a is called the shape parameter
b is called the scale parameter
The exponential distribution is a special case of the gamma with a =1.
The gamma distribution is used as a probability model for the time or space before the ath event in a Poisson process where events occur at the rate b=1/l.
Gamma calculator

12. Back to the clunker car
Recall that my car breaks down once a week on average. If the breakdowns occur as events in a Poisson process, then what is the probability less than a week passes before my first breakdown? Gamma or Poisson?
Gamma Calculator

13. Pipe example
Defects along a piece of pipe occur as events in a Poisson process with an average of 2 defects every 10 feet. What is the probability that the third defect will occur at least 20 feet from the beginning of the pipe?
Gamma Calculator

14. Normal distribution
The normal distribution, N(m,s2), has a pdf given by
The normal distribution is always bell shaped.
The normal distribution is defined in terms of its mean and variance (standard deviation).
Normal calculator

15. Weight gain example
The weight gain associated with an antidepressant is normally distributed with a mean of 6 lbs and a standard deviation of 3 lbs.
What is the probability of weight gain?
What is the probability of gaining between 0 and 12 lbs?
Normal Calculator

16. Standard normal distribution
If X has a N(m,s2) distribution, then Z=(X-m)/s has a standard normal distribution, N(0,1).
The standard normal is an important reference distribution.
P(X ≤ x) = P(Z ≤ (x-m)/s) = F((x-m)/s)
The cdf of a standard normal, F(z), is tabled in many textbooks
Standardized values, (x-m)/s, indicate how far in standard deviations the value x is from m
For any normal distribution, probabilities can be phrased in terms of standardized values

17. Empirical rule What is the probability Normal Calculator
a normal falls within one standard deviation of the mean?
a normal falls within two standard deviations of the mean?
a normal falls within three standard deviations of the mean?
Normal Calculator

18. Back to the weight gain example
Recall m=6 and s=3.
Using the empirical rule, answer the following questions:
What is the probability of weight loss?
What is the probability of a weight gain between 0 and 12 pounds?

19. Normal approximations
Normal approximation to Binomial
Normal approximation to Poisson

20. Do my data look normal?
In StatCrunch, a quantile-quantile plot (QQ plot) plots ordered data values versus quantiles of a standard normal distribution.
If the data are from a normal distribution, the points should lie approximately on a straight line.
Concentration data

21. Other distributions
The Weibull distribution and the log normal distribution are used to model failure times.
The beta distribution is used to model proportions.
There are many other distributions out there.
Choose the one that serves as the best probability model for your setting.